Thanks for all the great comments, and Facebook feature!!
To answer a couple questions:
My husband just packed the pieces of corrugated paper
so tight that they don't budge at all. (Though I imagine that once Gus gets a little stronger, they might need to be glued down. Thought at the moment, I can't even really get them out!)
I hot-glued all the batting to the insides/edges of the box, then hot-glued the material to the outside bottom and draped in. It is completely covered, but I also made a fluffier, removable pillow to stick in the bottom in case washing is needed. (It is also bigger and kinda folded in so it can come out and he will still have a place to lay if/when he gets too big for the box bed!)
I'll have to try the rolled up and with catnip idea. Gus would probably enjoy that!
And we like his name because it gives us a lot of options and variations, plus, he's got little white tap shoes on his paws!